MHB Is the Inequality 1/(log₂π)+1/(log₅π)>2 True? A Scientific Investigation

[anemone]

Prove $\dfrac{1}{\log_2 \pi}+\dfrac{1}{\log_5 \pi}>2 $.

 

[kaliprasad]

Spoiler

$\frac{1}{\log_2\pi } +\frac{1}{\log_5\pi } $
= $\log_{\pi} 2 +\log_{\pi} 5$ using $\log_a b * \log_b a = 1$
= $\log_{\pi} 10$

Now $\pi = 3.14 < 3.15$
so $\pi^2 < 31.5^2$ or $\pi^2 < 992.25$ as $31 * 32 = 992$
so $ 10 > \pi^2$
so we have
$\log_{\pi} 10 > 2$ and hence the result